Tensor K-matrices for quantum symmetric pairs
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arXiv
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| Format: | Preprint |
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2024
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| author | Appel, Andrea Vlaar, Bart |
| author_facet | Appel, Andrea Vlaar, Bart |
| contents | Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, $U_q(\mathfrak{g})$ its quantum group, and $U_q(\mathfrak{k}) \subset U_q(\mathfrak{g})$ a quantum symmetric pair subalgebra determined by a Lie algebra automorphism $θ$. We introduce a category $W_θ$ of weight $U_q(\mathfrak{k})$-modules, which is acted on by the category of weight $U_q(\mathfrak{g})$-modules via tensor products. We construct a universal tensor K-matrix $\mathbb{K}$ (that is, a solution of a reflection equation) in a completion of $U_q(\mathfrak{k}) \otimes U_q(\mathfrak{g})$. This yields a natural operator on any tensor product $M \otimes V$, where $M\in W_θ$ and $V\in {O}_θ$, that is $V$ is a $U_q(\mathfrak{g})$-module in category ${O}$ satisfying an integrability property determined by $θ$. Canonically, $W_θ$ is equipped with a structure of a bimodule category over ${O}_θ$ and the action of $\mathbb{K}$ is encoded by a new categorical structure, which we call a boundary structure on $W_θ$. This generalizes a result of Kolb which describes a braided module structure on finite-dimensional $U_q(\mathfrak{k})$-modules when $\mathfrak{g}$ is finite-dimensional. We also consider our construction in the case of the category ${C}$ of finite-dimensional modules over a quantum affine algebra, providing the most comprehensive universal framework to date for large families of solutions of parameter-dependent reflection equations. In this case the tensor K-matrix gives rise to a formal Laurent series with a well-defined action on tensor products of any module in $W_θ$ and any module in ${C}$. This series can be normalized to an operator-valued rational function, which we call trigonometric tensor K-matrix, if both factors in the tensor product are in ${C}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_16676 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tensor K-matrices for quantum symmetric pairs Appel, Andrea Vlaar, Bart Representation Theory 81R50, 16T25, 17B37, 81R12 Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, $U_q(\mathfrak{g})$ its quantum group, and $U_q(\mathfrak{k}) \subset U_q(\mathfrak{g})$ a quantum symmetric pair subalgebra determined by a Lie algebra automorphism $θ$. We introduce a category $W_θ$ of weight $U_q(\mathfrak{k})$-modules, which is acted on by the category of weight $U_q(\mathfrak{g})$-modules via tensor products. We construct a universal tensor K-matrix $\mathbb{K}$ (that is, a solution of a reflection equation) in a completion of $U_q(\mathfrak{k}) \otimes U_q(\mathfrak{g})$. This yields a natural operator on any tensor product $M \otimes V$, where $M\in W_θ$ and $V\in {O}_θ$, that is $V$ is a $U_q(\mathfrak{g})$-module in category ${O}$ satisfying an integrability property determined by $θ$. Canonically, $W_θ$ is equipped with a structure of a bimodule category over ${O}_θ$ and the action of $\mathbb{K}$ is encoded by a new categorical structure, which we call a boundary structure on $W_θ$. This generalizes a result of Kolb which describes a braided module structure on finite-dimensional $U_q(\mathfrak{k})$-modules when $\mathfrak{g}$ is finite-dimensional. We also consider our construction in the case of the category ${C}$ of finite-dimensional modules over a quantum affine algebra, providing the most comprehensive universal framework to date for large families of solutions of parameter-dependent reflection equations. In this case the tensor K-matrix gives rise to a formal Laurent series with a well-defined action on tensor products of any module in $W_θ$ and any module in ${C}$. This series can be normalized to an operator-valued rational function, which we call trigonometric tensor K-matrix, if both factors in the tensor product are in ${C}$. |
| title | Tensor K-matrices for quantum symmetric pairs |
| topic | Representation Theory 81R50, 16T25, 17B37, 81R12 |
| url | https://arxiv.org/abs/2402.16676 |